How do you connect two edge loops with a face between?
There are several way of doing this.  


The easiest is to use the vertex connect path tool J which will make a new edge between the two vertices, splitting the face into two.
The next best thing would be to use the knife tool K and just cut in new edge between the two vertices. Again that will take care of the faces.
To use the fill tool F you will first have to delete the N-gon in the middle. Then create two faces.


In this gif I go through the three methods listed above. First using the vertex connect tool, then the knife tool, and last creating the two faces manually with the fill tool.